Question

find the length of the missing side of the triangle. for any answers with a square root, simplify and write your answer in the form sqrt(_) with no spaces.
\\(x = \\)

Explanation:

⚡ Using what you learned: the pythagorean theorem and its converse

Step 1: Identify the sides of the right triangle

The given triangle is a right triangle.

• The legs are \( a = \sqrt{101} \) and \( b = 12 \).
• The hypotenuse (the side opposite the right angle) is \( c = x \).

Step 2: Apply the Pythagorean theorem

$$ a^2 + b^2 = c^2 $$

Substitute the known values into the formula:

$$ (\sqrt{101})^2 + 12^2 = x^2 $$

$$ 101 + 144 = x^2 $$

$$ 245 = x^2 $$

Step 3: Solve for x and simplify

$$ x = \sqrt{245} $$

To simplify the radical, find the largest perfect square factor of \( 245 \):

$$ 245 = 49 \times 5 $$

$$ x = \sqrt{49 \times 5} = 7\sqrt{5} $$

Since the question asks to write the answer in the form sqrt(__) with no spaces if it contains a square root, we write the fully simplified radical form as:

$$ 7\sqrt{5} $$

Answer:

7sqrt(5)